STOCKHOLM—Felix Kjellberg doesn't play a very convincing megastar. The seemingly modest Swede avoids the spotlight, has no entourage and is uncomfortable with success.

But put Mr. Kjellberg in front of a Web-camera and he transforms into PewDiePie, by far YouTube's biggest draw. He has built a base of 27 million subscribers using a decidedly unorthodox approach to playing video and mobile games.